Projective planes and Hadamard designs (Q5929835)

Let \(\pi\) be a projective plane of order \(n\geq 8\). If \(n\equiv 0\) (mod 4) and \(\pi\) has an elation group of order \({1\over 2} n\) with common center \(P_0\) and common axis \(l_0\), then it is shown in the paper under review that there exists a Hadamard design with parameters \((n-1, {1 \over 2} n-1, {1 \over 4} n-1)\). Similarly, it is shown that if \(\pi\) is a projective plane of order \(n \geq 7\) with \(n\equiv 3\) (mod 4) such that \(\pi\) admits a homology group of order \({1 \over 2}(n-1)\) with common center \(P_0\) and common axis \(l_0\), then there exists a Hadamard design with parameters \((n, {1\over 2} (n-1),{1\over 4} (n-3))\). In both cases the idea is to number the point and line orbits appropriately, use this numbering scheme to create an incidence matrix, and then use computations in the group ring over \(\mathbb Z \) to show that the incidence matrix represents a Hadamard design. The Hadamard designs with parameters \((n-1, {1\over 2} n-1,{1 \over 4} n-1)\) which arise from translation planes are shown to correspond to Singer difference sets. It would be interesting to study the Hadamard designs arising from non-translation planes satisfying the elation group condition, such as the derived dualized Lüneburg-Tits planes, but this was not addressed in the paper. NEWLINENEWLINENEWLINEPartial converses of the above theorems are given, and some applications are presented. For instance, it is shown that there cannot exist a projective plane of order 12 with an elation of order 6, and every projective plane of order 11 with a homology of order 5 is necessarily desarguesian.